The challenging conjecture of H. S. Snevily [Am. Math. Mon. 106, No. 6, 584-585 (1999)] on subsets in Abelian groups of odd order which is described as follows has attracted much attention: let \(A=\{a_1,\dots,a_k\}\) and \(B=\{b_1,\dots,b_k\}\) be two \(k\)-subsets of Abelian group \(G\) of odd order (written multiplicatively), then there is a permutation \(\pi\in S_k\) such that \(a_1b_{\pi(1)},\dots,a_kb_{\pi(k)}\) are distinct. N. Alon [Isr. J. Math. 117, 125-130 (2000; Zbl 1047.11019)] proved the conjecture when the order of the group is an odd prime, and in 2001, S. Dasgupta, G. Károlyi, O. Serra and B. Szegedy, [Isr. J. Math. 126, 17-28 (2001; Zbl 1011.05014)], proved it for any cyclic group of odd order. In this paper, the author proves the conjecture ingeniously and briefly with the skills of linear algebra.